Planar Sensor Array

ABSTRACT

A method ( 500 ) for constructing a three-dimensional (3D) wave field representation of a 3D wave field using a two-dimensional (2D) sensor array ( 110 ), said method comprising: acquiring 3D wave field signals using a 2D array ( 110 ) of sensors ( 340, 350 ), said 2D array ( 110 ) of sensors ( 340, 350 ) comprising omnidirectional sensors ( 340 ) and first order sensors ( 350 ) arranged in a 2D plane; digitizing said acquired 3D wave field signals; computing even coefficients of spherical harmonics dependent upon said digitized 3D wave field signals acquired by said omnidirectional sensors ( 340 ); computing odd coefficients of said spherical harmonics dependent upon said digitized 3D wave field signals acquired by said first order sensors ( 350 ); and constructing a 3D wave field representation dependent upon said computed even and odd coefficients for said acquired 3D wave field signals.

TECHNICAL FIELD

The present invention relates generally to the field of signalprocessing and, in particular, to acquiring three-dimensional (3D) wavefield signals using a two-dimensional (2D) sensor array and constructinga 3D wave field representation of the acquired, actual 3D wave fieldsignals.

BACKGROUND

In the field of signal processing, it is desirable to obtain a 3D wavefield mathematical representation of the actual 3D wave field signals assuch a representation enables an accurate analysis of the 3D wave field.One such mathematical representation is the 3D wave field sphericalharmonic decomposition.

3D wave field signals in a spherical coordinate system (r, θ, Φ) can bemathematically represented by equation 1 as an infinite sum of sphericalharmonics:

$\begin{matrix}{{P\left( {r,\theta,\varphi,k} \right)} = {\sum\limits_{n = 0}^{\infty}\; {\sum\limits_{m = {- n}}^{n}\; {C_{nm}{j_{n}({kr})}{_{n{m}}\left( {\cos \mspace{11mu} \theta} \right)}{E_{m}(\varphi)}}}}} & \left( {{eq}.\mspace{14mu} 1} \right)\end{matrix}$

where C_(nm) is the coefficient, j_(N)(kr) is the spherical Besselfunction,Y_(nm)=P_(n|m|)(cos θ)E_(m)(φ) is a representation of the sphericalharmonics,

$\begin{matrix}{{_{n{m}}\left( {\cos \mspace{11mu} \theta} \right)} = {\sqrt{\frac{\left( {{2\; n} + 1} \right)}{4\; \pi}\frac{\left( {n - m} \right)!}{\left( {n + m} \right)!}}{P_{n{m}}\left( {\cos \mspace{11mu} \theta} \right)}}} & \left( {{eq}.\mspace{14mu} 2} \right)\end{matrix}$

is the normalized Associated Legendre function, and E_(m)(φ)=(1/√{squareroot over (2π)})e^(2mφ) is the normalized exponential function. Thenormalized exponential function represents spherical waves in the φdirection, while the normalized Associated Legendre function representsspherical waves in the θ direction.

The spherical harmonics are orthonormal, therefore satisfying:

∫_(S) ₂ Y _(n′m′)(Ω)Y _(nm)*(Ω)d(Ω)=τ_(n-n′)τ_(m-m′)  (eq. 3)

where Y_(nm)=

_(n|m|)(cos θ)E_(m)(φ) is a representation of the spherical harmonics.

FIG. 10 shows a plot of the spherical harmonics of order 0 to 3, whichshows that the odd-modes 1010 of the spherical harmonics are zero at

${\theta = \frac{\pi}{2}},$

and that the even-modes 1020 of the spherical harmonics are non-zero at

$\theta = {\frac{\pi}{2}.}$

FIG. 10 also shows the spherical coordinate system corresponding to thespherical harmonics. The even-modes 1020 are only partially marked inFIG. 10 to avoid cluttering the figure. According to the sphericalharmonics, only even-modes 1020 are observable on the x-y plane (i.e.,

$\theta = \frac{\pi}{2}$

plane). That is, odd modes 1010 are undetectable on the x-y plane.Therefore, sensors need to be placed at different vertical altitudes toacquire the 3D wave field signals in order to fully produce themathematical representation of the 3D wave field spherical harmonicsdecomposition.

One type of array configuration fulfilling the above requirement is thespherical array. The geometry of the spherical array coincides with thespherical harmonics, which makes the 3D wave field signals acquired bythe spherical array to be suitable for generating 3D wave fieldspherical harmonics decomposition. There are two models of the sphericalarray configuration: the open sphere model (where the sensors are placedin open space) and the rigid sphere model (where the sensors are placedon the surface of a rigid sphere).

A problem however exists with the spherical array in that the arraycould be ill-conditioned numerically, due to nulls in the sphericalBessel Functions. This problem results in the acquired 3D wave signalsbeing highly sensitivity to the diameter of the spherical array. Inaddition, placement of sensors on the spherical array follows a strictrule of orthogonality of the spherical harmonics, which limits theflexibility of the array configuration (especially in terms of sensorquantity). Further, the spherical shape of the array poses difficultiesin regard to implementation as well as being impractical.

Another limitation of the spherical array is the narrow frequency band,due to the nature of the spherical Bessel function. The spherical arrayis therefore unable to process 3D wave field signals for certain ordersof the spherical harmonics. Design of the spherical array must beperformed carefully so that the active spherical Bessel functions arenon-zero at the spherical array's radius, for the target frequency band.

Thus, a need exists to provide more practical array configurations.

SUMMARY

Disclosed are arrangements which seek to address the above problems byusing a 2D sensor array to acquire the 3D wave field signals andconstruct the 3D wave field representation from the acquired 3D wavefield signals.

According to a first aspect of the present disclosure, there is provideda method for constructing a three-dimensional (3D) wave fieldrepresentation of a 3D wave field using a two-dimensional (2D) sensorarray, said method comprising: acquiring 3D wave field signals using a2D array of sensors, said 2D array of sensors comprising omnidirectionalsensors and first order sensors arranged in a 2D plane; digitizing saidacquired 3D wave field signals; computing even coefficients of sphericalharmonics dependent upon said digitized 3D wave field signals acquiredby said omnidirectional sensors; computing odd coefficients of saidspherical harmonics dependent upon said digitized 3D wave field signalsacquired by said first order sensors; and constructing a 3D wave fieldrepresentation dependent upon said computed even and odd coefficientsfor said acquired 3D wave field signals.

According to another aspect of the present disclosure, there is provideda computer program product including a computer readable medium havingrecorded thereon a computer program for implementing the methoddescribed above.

Other aspects of the invention are also disclosed.

BRIEF DESCRIPTION OF THE DRAWINGS

At least one embodiment of the present invention will now be describedwith reference to the drawings, in which:

FIG. 1 shows a block diagram of a system using a two-dimensional (2D)array to acquire three-dimensional (3D) wave field signals and constructthe 3D wave field representation according to the present disclosure;

FIGS. 2A and 2B form a schematic block diagram of a general purposecomputer system upon which arrangements described can be practiced;

FIGS. 3A and 3B show configurations of the 2D sensor array of the systemof FIG. 1;

FIG. 3C is a flow diagram of a method for designing the 2D sensor array;

FIGS. 4A and 4B show block diagrams of implementations of the system ofFIG. 1;

FIG. 4C shows plots (a) and (c) of sound field captured and plots (b)and (d) of reconstructed sound field by the system of FIG. 1A;

FIG. 5 is a flow diagram of a method for constructing athree-dimensional (3D) wave field representation of a 3D wave fieldusing a two-dimensional (2D) sensor array of FIG. 1;

FIG. 6A is a block diagram of the system of FIG. 1 being implemented inan active noise cancellation application according to the presentdisclosure;

FIG. 6B is an error plot of the active noise cancellation system of FIG.6A;

FIG. 7 is a block diagram of the system of FIG. 1 being implemented in abeamforming application according to the present disclosure;

FIG. 8 is a block diagram of the system of FIG. 1 being implemented in adirection of arrival estimation application according to the presentdisclosure;

FIG. 9 is a block diagram of the system of FIG. 1 being implemented in a3D sound field reproduction application according to the presentdisclosure; and

FIG. 10 shows a representation of the spherical harmonics of the thirdorder.

DETAILED DESCRIPTION

Where reference is made in any one or more of the accompanying drawingsto steps and/or features, which have the same reference numerals, thosesteps and/or features have for the purposes of this description the samefunction(s) or operation(s), unless the contrary intention appears.

Disclosed is an arrangement of a 2D sensor array having omnidirectionaland first order (i.e., directional) sensors to acquire 3D wave fieldsignals so that a mathematical representation of the acquired 3D wavefield signals can be constructed. The omnidirectional sensors functionto acquire the even-modes (i.e., the horizontal wave field) components,whilst the first order sensors measure the odd-modes (i.e., the verticalwave field) components. The 3D wave field representation can then beconstructed from the acquired even- and odd-modes components.

FIG. 1 shows a system 100 for acquiring 3D wave field signals andconstructing a 3D wave field representation of the acquired 3D wavefield signals. The system 100 includes a 2D sensor array 110 and acomputer system 120. The sensor array 110 is arranged in a plurality ofconcentric circular arrays, where each concentric circular array has aradius (different to the other concentric circular arrays) and adiscrete number of sensors. The term “sensors” is used hereinafter forconsistency, but the terms “antennas”, “microphones”, and “hydrophones”are equally applicable depending on the circumstances. According to thetype of sensor used, the acquired 3D wave field signals may be any oneof acoustic wave field signals, radio frequency wave field signals, andmicrowave wave field signals. The arrangement of the 2D sensor array 110will be described in detail in relation to FIGS. 3 to 5 below.

The computer system 120 has an array processing module 130 and anapplication module 140. The array processing module 130 processes 3Dwave field signals acquired by the 2D sensor array 110 and generates 3Dwave field spherical harmonics decompositions (i.e., even- and odd-modescoefficients) from the acquired 3D wave field signals. The applicationmodule 140 uses the generated 3D wave field decompositions in any one ofthese applications: active noise cancellation, beamforming, direction ofarrival estimation, and 3D sound field recording/reproduction.

The computer system 120 can be a general purpose computer system or aspecial purpose computer dedicated to execute the coded of the arrayprocessing module 130 and the application module 140. The method foracquiring the 3D wave field signals and generating the 3D wave fieldspherical harmonics decomposition by the array processing module 130will be discussed hereinafter in relation to FIG. 5. The implementationsof the application module 140 in using the 3D wave field decompositiongenerated by the array processing module 130 will be describedhereinafter in relation to FIGS. 6 to 9.

Computer Description

FIGS. 2A and 2B depict a general-purpose computer system 120, upon whichthe various arrangements described can be practiced.

As seen in FIG. 2A, the computer system 120 includes: a computer module201; input devices such as a keyboard 202, a mouse pointer device 203, ascanner 226, a camera 227, and a microphone 280; and output devicesincluding a printer 215, a display device 214 and loudspeakers 217. Anexternal Modulator-Demodulator (Modem) transceiver device 216 may beused by the computer module 201 for communicating to and from acommunications network 220 via a connection 221. The communicationsnetwork 220 may be a wide-area network (WAN), such as the Internet, acellular telecommunications network, or a private WAN. Where theconnection 221 is a telephone line, the modem 216 may be a traditional“dial-up” modem. Alternatively, where the connection 221 is a highcapacity (e.g., cable) connection, the modem 216 may be a broadbandmodem. A wireless modem may also be used for wireless connection to thecommunications network 220.

The computer module 201 typically includes at least one processor unit205, and a memory unit 206. For example, the memory unit 206 may havesemiconductor random access memory (RAM) and semiconductor read onlymemory (ROM). The computer module 201 also includes an number ofinput/output (I/O) interfaces including: an audio-video interface 207that couples to the video display 214, loudspeakers 217 and microphone280; an I/O interface 213 that couples to the keyboard 202, mouse 203,scanner 226, camera 227, the 2D sensor array 110, and optionally ajoystick or other human interface device (not illustrated); and aninterface 208 for the external modem 216 and printer 215. In someimplementations, the modem 216 may be incorporated within the computermodule 201, for example within the interface 208. The computer module201 also has a local network interface 211, which permits coupling ofthe computer system 120 via a connection 223 to a local-areacommunications network 222, known as a Local Area Network (LAN). Asillustrated in FIG. 2A, the local communications network 222 may alsocouple to the wide network 220 via a connection 224, which wouldtypically include a so-called “firewall” device or device of similarfunctionality. The local network interface 211 may comprise an Ethernetcircuit card, a Bluetooth wireless arrangement or an IEEE 802.11wireless arrangement; however, numerous other types of interfaces may bepracticed for the interface 211.

The I/O interfaces 208 and 213 may afford either or both of serial andparallel connectivity, the former typically being implemented accordingto the Universal Serial Bus (USB) standards and having corresponding USBconnectors (not illustrated). Storage devices 209 are provided andtypically include a hard disk drive (HDD) 210. Other storage devicessuch as a floppy disk drive and a magnetic tape drive (not illustrated)may also be used. An optical disk drive 212 is typically provided to actas a non-volatile source of data. Portable memory devices, such opticaldisks (e.g., CD-ROM, DVD, Blu-ray Disc™), USB-RAM, portable, externalhard drives, and floppy disks, for example, may be used as appropriatesources of data to the system 120.

The components 205 to 213 of the computer module 201 typicallycommunicate via an interconnected bus 204 and in a manner that resultsin a conventional mode of operation of the computer system 120 known tothose in the relevant art. For example, the processor 205 is coupled tothe system bus 204 using a connection 218. Likewise, the memory 206 andoptical disk drive 212 are coupled to the system bus 204 by connections219. Examples of computers on which the described arrangements can bepractised include IBM-PC's and compatibles, Sun Sparcstations, AppleMac™ or a like computer systems.

The method of processing acquired 3D wave field signals to generate 3Dwave field decomposition of the spherical harmonics may be implementedusing the computer system 120 wherein the processes of FIG. 5, to bedescribed, may be implemented as one or more software applicationprograms 233 executable within the computer system 120. In particular,the steps of the method of generating 3D wave field composition of thespherical harmonics from the 2D sensor array 110 are effected byinstructions 231 (see FIG. 2B) in the software 233 that are carried outwithin the computer system 120. The software instructions 231 may beformed as one or more code modules, each for performing one or moreparticular tasks. The software may also be divided into two separateparts, in which a first part and the corresponding code module (i.e.,modules 130 and 140) acquires the 3D wave field signals from the 2Dsensor array 110 and performs the 3D wave field decomposition of thespherical harmonics and a second part and the corresponding code modulesmanage a user interface between the first part and the user.

The software may be stored in a computer readable medium, including thestorage devices described below, for example. The software is loadedinto the computer system 120 from the computer readable medium, and thenexecuted by the computer system 120. A computer readable medium havingsuch software or computer program recorded on the computer readablemedium is a computer program product. The use of the computer programproduct in the computer system 120 preferably effects an advantageousapparatus for generating 3D wave field decomposition of the sphericalharmonics from 3D wave field signals acquired by the 2D sensor array110.

The software 233 is typically stored in the HDD 210 or the memory 206.The software is loaded into the computer system 120 from a computerreadable medium, and executed by the computer system 120. Thus, forexample, the software 233 may be stored on an optically readable diskstorage medium (e.g., CD-ROM) 225 that is read by the optical disk drive212. A computer readable medium having such software or computer programrecorded on it is a computer program product. The use of the computerprogram product in the computer system 120 preferably effects anapparatus for generating 3D wave field decomposition of the sphericalharmonics from 3D wave field signals acquired by the 2D sensor array110.

In some instances, the application programs 233 may be supplied to theuser encoded on one or more CD-ROMs 225 and read via the correspondingdrive 212, or alternatively may be read by the user from the networks220 or 222. Still further, the software can also be loaded into thecomputer system 120 from other computer readable media. Computerreadable storage media refers to any non-transitory tangible storagemedium that provides recorded instructions and/or data to the computersystem 120 for execution and/or processing. Examples of such storagemedia include floppy disks, magnetic tape, CD-ROM, DVD, Blu-Ray™ Disc, ahard disk drive, a ROM or integrated circuit, USB memory, amagneto-optical disk, or a computer readable card such as a PCMCIA cardand the like, whether or not such devices are internal or external ofthe computer module 201. Examples of transitory or non-tangible computerreadable transmission media that may also participate in the provisionof software, application programs, instructions and/or data to thecomputer module 201 include radio or infra-red transmission channels aswell as a network connection to another computer or networked device,and the Internet or Intranets including e-mail transmissions andinformation recorded on Websites and the like.

The second part of the application programs 233 and the correspondingcode modules mentioned above may be executed to implement one or moregraphical user interfaces (GUIs) to be rendered or otherwise representedupon the display 214. Through manipulation of typically the keyboard 202and the mouse 203, a user of the computer system 120 and the applicationmay manipulate the interface in a functionally adaptable manner toprovide controlling commands and/or input to the applications associatedwith the GUI(s). Other forms of functionally adaptable user interfacesmay also be implemented, such as an audio interface utilizing speechprompts output via the loudspeakers 217 and user voice commands inputvia the microphone 280.

FIG. 2B is a detailed schematic block diagram of the processor 205 and a“memory” 234. The memory 234 represents a logical aggregation of all thememory modules (including the HDD 209 and semiconductor memory 206) thatcan be accessed by the computer module 201 in FIG. 2A.

When the computer module 201 is initially powered up, a power-onself-test (POST) program 250 executes. The POST program 250 is typicallystored in a ROM 249 of the semiconductor memory 206 of FIG. 2A. Ahardware device such as the ROM 249 storing software is sometimesreferred to as firmware. The POST program 250 examines hardware withinthe computer module 201 to ensure proper functioning and typicallychecks the processor 205, the memory 234 (209, 206), and a basicinput-output systems software (BIOS) module 251, also typically storedin the ROM 249, for correct operation. Once the POST program 250 has runsuccessfully, the BIOS 251 activates the hard disk drive 210 of FIG. 2A.Activation of the hard disk drive 210 causes a bootstrap loader program252 that is resident on the hard disk drive 210 to execute via theprocessor 205. This loads an operating system 253 into the RAM memory206, upon which the operating system 253 commences operation. Theoperating system 253 is a system level application, executable by theprocessor 205, to fulfil various high level functions, includingprocessor management, memory management, device management, storagemanagement, software application interface, and generic user interface.

The operating system 253 manages the memory 234 (209, 206) to ensurethat each process or application running on the computer module 201 hassufficient memory in which to execute without colliding with memoryallocated to another process. Furthermore, the different types of memoryavailable in the system 120 of FIG. 2A must be used properly so thateach process can run effectively. Accordingly, the aggregated memory 234is not intended to illustrate how particular segments of memory areallocated (unless otherwise stated), but rather to provide a generalview of the memory accessible by the computer system 120 and how such isused.

As shown in FIG. 2B, the processor 205 includes a number of functionalmodules including a control unit 239, an arithmetic logic unit (ALU)240, and a local or internal memory 248, sometimes called a cachememory. The cache memory 248 typically includes a number of storageregisters 244-246 in a register section. One or more internal busses 241functionally interconnect these functional modules. The processor 205typically also has one or more interfaces 242 for communicating withexternal devices via the system bus 204, using a connection 218. Thememory 234 is coupled to the bus 204 using a connection 219.

The application program 233 includes a sequence of instructions 231 thatmay include conditional branch and loop instructions. The program 233may also include data 232 which is used in execution of the program 233.The instructions 231 and the data 232 are stored in memory locations228, 229, 230 and 235, 236, 237, respectively. Depending upon therelative size of the instructions 231 and the memory locations 228-230,a particular instruction may be stored in a single memory location asdepicted by the instruction shown in the memory location 230.Alternately, an instruction may be segmented into a number of parts eachof which is stored in a separate memory location, as depicted by theinstruction segments shown in the memory locations 228 and 229.

In general, the processor 205 is given a set of instructions which areexecuted therein. The processor 1105 waits for a subsequent input, towhich the processor 205 reacts to by executing another set ofinstructions. Each input may be provided from one or more of a number ofsources, including data generated by one or more of the input devices202, 203, data received from an external source across one of thenetworks 220, 202, data retrieved from one of the storage devices 206,209 or data retrieved from a storage medium 225 inserted into thecorresponding reader 212, all depicted in FIG. 2A. The execution of aset of the instructions may in some cases result in output of data.Execution may also involve storing data or variables to the memory 234.

The disclosed arrangements use input variables 254, which are stored inthe memory 234 in corresponding memory locations 255, 256, 257. Thedisclosed arrangements produce output variables 261, which are stored inthe memory 234 in corresponding memory locations 262, 263, 264.Intermediate variables 258 may be stored in memory locations 259, 260,266 and 267.

Referring to the processor 205 of FIG. 2B, the registers 244, 245, 246,the arithmetic logic unit (ALU) 240, and the control unit 239 worktogether to perform sequences of micro-operations needed to perform“fetch, decode, and execute” cycles for every instruction in theinstruction set making up the program 233. Each fetch, decode, andexecute cycle comprises:

a fetch operation, which fetches or reads an instruction 231 from amemory location 228, 229, 230;

a decode operation in which the control unit 239 determines whichinstruction has been fetched; and

an execute operation in which the control unit 239 and/or the ALU 240execute the instruction.

Thereafter, a further fetch, decode, and execute cycle for the nextinstruction may be executed. Similarly, a store cycle may be performedby which the control unit 239 stores or writes a value to a memorylocation 232.

Each step or sub-process in the processes of FIG. 5 is associated withone or more segments of the program 233 and is performed by the registersection 244, 245, 247, the ALU 240, and the control unit 239 in theprocessor 205 working together to perform the fetch, decode, and executecycles for every instruction in the instruction set for the notedsegments of the program 233.

The method of acquiring 3D wave field signals from the 2D sensor array110 and generating 3D wave field decomposition of the sphericalharmonics based on the acquired 3D wave field signals may alternativelybe implemented in dedicated hardware such as one or more integratedcircuits performing the functions or sub functions of the method of FIG.5. Such dedicated hardware may include a field-programmable gate array,graphic processors, digital signal processors, or one or moremicroprocessors and associated memories.

As mentioned in paragraph [0005] above, the odd-modes 1010 of the 3Dwave field spherical harmonics are undetectable when the sensor array110 is located on the x-y plane. However, as will be discussedhereinafter, the odd-modes spherical harmonics of the 3D wave field canbe acquired by sensors having directional reception pattern (i.e., firstorder sensor) perpendicular to the x-y plane. Thus, the 2D sensor array110 is capable of acquiring both the even- and odd-modes sphericalharmonics using a combination of omnidirectional sensors (to obtain theeven-modes spherical harmonics coefficients) and first order sensorswith directional reception pattern perpendicular to the x-y plane (toobtain the odd-modes spherical harmonics coefficients).

FIGS. 3A and 3B show two example configurations of the 2D sensor array110 for detecting the 3D wave field signals. FIG. 3A shows a 2D sensorarray 110A having first order sensors 350 arranged in a plurality ofconcentric circular arrays (310A, 310B, 310C, 310D, and 310E) withcorresponding radius (320A, 320B, 320C, 320D, and 320E, respectively).The concentric circular arrays (310A, 310B, 310C, 310D, and 310E) arecollectively referred to hereinafter as the circular arrays 310.However, the circular array 310N or the radius 320N is used hereinafterwhen referring to the largest of the circular array 310. The radius 320Nis also used hereinafter to refer to the size of the 2D sensor array110. Each concentric circular array 310 includes a number of sensors350.

In the array 110, each of the first order sensors 350 is formed by twoomnidirectional sensors 340 placed in close proximity to each other. Thedistance between the two omnidirectional sensors 340 in forming a firstorder sensor 350 is small compared to the array radius 320N. The firstorder (i.e., directional) sensor 350 has an opposite, bi-directionalreception pattern and is oriented such that the bi-directional receptionpattern is perpendicular to the plane of the array 110A. The output ofthe first order sensor 350 is the differential between the outputs ofthe omnidirectional sensors 340 forming the first order sensor 350.

One of the omnidirectional sensors 340, which is a part of the firstorder sensor 350, functions to receive the even-modes sphericalharmonics. The first order sensor 350, using both of the omnidirectionalsensors 340 that form the sensor 350, functions to receive the odd-modesspherical harmonics. The functionalities of the array 110A will bediscussed further in relation to FIG. 4A.

The number of first order sensors 350 on each circular array 310 isgiven by equation:

$\begin{matrix}{N_{X} = {{{2\; N} + 1} = {{{2\left\lceil \frac{e\; k\; R}{2} \right\rceil} + 1} = {{{2\left\lceil \frac{e\; \pi \; R}{\lambda} \right\rceil} + 1} = {{2\left\lceil \frac{e\; \pi \; f\; R}{c} \right\rceil} + 1}}}}} & \left( {{eq}.\mspace{14mu} 4} \right)\end{matrix}$

where N_(X) is the number of sensors 350 on each circular array 310, Nis the maximum observable spherical harmonic order, k is the wave numberof the design frequency, R is the radius 320 of the circular array 310,and c is the speed of the wave. For audio applications, c=340 m/s; forRF applications, c=300,000,000 m/s. For example, a circular array 310 of0.2 m radius that is designed to receive 900 MHz RF signal would have 13sensors (e.g., antennas). In another example, a circular array 310 of0.4 m radius that is designed to receive audio signals up to 1500 Hzwould have 33 sensors (e.g., microphones).

FIG. 3B shows an array 110B having omnidirectional sensors 340 and firstorder sensors 350 arranged in a plurality of concentric circular arrays(310A, 310B, 310C, 310D, 310E, and 310F) with corresponding radius(320A, 320B, 320C, 320D, 320E, and 320F, respectively). Each concentriccircular array 310 has either the omnidirectional sensors 340 or thefirst order sensors 350. The array 1100B is arranged such that aconcentric circular array (i.e., 310A, 310C, and 310E) withomnidirectional sensors 340 is alternated with a concentric circulararray (e.g., 310B, 310D, and 310F) with first order sensors 350. Theconcentric circular arrays (310A, 310B, 310C, 310D, 310E, and 310F) arecollectively referred to hereinafter as the circular arrays 310, similarto the circular arrays 310 of the array 110A.

In the array 110B, cardioid sensors form the first order sensors 350.Thus, the omnidirectional sensors 340 function to receive the even-modesspherical harmonics, whilst the first order sensors 350 function toreceive the odd-modes spherical harmonics. The functionalities of thearray 110B will be discussed further in relation to FIG. 4B. However,the first order sensor 350 may also be formed by two omnidirectionalsensors 340, as implemented in the array 11A of FIG. 3A.

FIG. 3C shows a flowchart of a method 300 for designing the 2D sensorarray 110. The method 300 commences at step 362 where the designparameters of the 2D sensor array 110 are selected. The designparameters include the maximum radius 320N, the highest wave number k tobe acquired, and the type of first order sensors 350 to be used. Forexample, to design a 2D sensor array 110 for active noise cancellationin a room of dimensions 3 m×3 m, the designer may select the maximumradius 320N to be 1.5 m so that the array 110 fits on the ceiling of theroom. The highest wave number k is selected depending on the highestfrequency to be acquired. A highest frequency of 850 Hz results in

${k = {k = {\frac{2\pi \; f}{c} = 15.7}}},$

where c=340 m/s for an audio application, for example. The method 300then proceeds to step 364.

In step 364, the maximum obtainable order of the wave field is computed.The maximum obtainable order of the wave field N is obtainable by usingequation

$N = {\left\lceil \frac{e\; k\; R}{2} \right\rceil.}$

The method 300 continues to step 366.

In step 366, the number of concentric circular arrays 310 is determined.Based on the maximum number of wave field order N, as computed in step364, the number of circular arrays 310 can be chosen. For the 2D sensorarray 110A, the total number of circular arrays 310 for the first ordersensors 350 is at least N_(first)=┌N/2┐. For the 2D sensor array 110B,the total number of circular arrays 310 for the omnidirectional sensors340 is N_(omni)=┌N/2┐, and the total number of circular arrays 310 forthe first order sensors 350 is N_(first)=N−N_(omni). The method 300 thenproceeds to step 368.

In step 368, the radius 320 for each concentric circular array 310 isdetermined. The radius 320 for each circular array 310 is chosen suchthat the circular arrays 310 are distributed so that the sphericalBessel zeros at the target frequency band are avoided. The sphericalBessel functions (shown in eq. 10 hereinafter) are used to determine theradius 320 of each circular array 310. For a given frequency k, thespherical Bessel function j_(n)(kr) becomes zero at certain radius 320.If a circular array 310 is placed at the radius 320 where the sphericalBessel function is zero, then the product of the spherical Besselfunction (i.e., C_(nm)j_(n)(kr)P_(nm)(cos θ)E_(m)(φ)) becomes zero, dueto the “Bessel zero”, which makes calculation of C_(nm) very difficult.Therefore, radius 320 where the “Bessel zeros” would occur should beavoided when designing the circular arrays 310. The method 300 thenproceeds to step 370.

In step 370, the number of sensors (340 and 350) is determined. Thenumber of sensors (340 and 350) on each circular array 310 is given byequation 4 above. The method 300 concludes after step 370.

FIG. 4A shows the implementation of the array 110A on the system 100. Asdiscussed in relation to FIG. 3A above, the first order sensor 350 ofthe array 110A is formed by two omnidirectional sensors (340A and 340B).The output of each of the omnidirectional sensors 340 is transmitted tothe computer system 120 which is then processed by the array processingmodule 130A, as the array processing module 130A is executed by theprocessor 205.

The array processing module 130A has fast Fourier transform (FFT)modules (430A, 430B), differential modules 440, even coefficients module410A, and odd coefficients module 420A. The FFT modules 430A and 430Bare collectively referred to hereinafter as the FFT modules 430. The FFTmodules 430 digitize the 3D wave signals acquired by the omnidirectionalsensor 340. The even coefficients module 410A processes the digitized 3Dwave field signals, acquired by an omnidirectional sensor 340B, tocalculate the even-mode spherical harmonic coefficients of the acquired3D wave field signals. The odd coefficients module 420A processes thedigitized 3D wave field signals, acquired by the first order sensors350, to calculate the even-mode spherical harmonic coefficients of theacquired 3D wave field signals.

To determine the even-modes spherical harmonics coefficients, the outputof the omnidirectional sensor 340B is transmitted to the FFT module 430Band the output of the FFT module 430B is transmitted to the evencoefficients module 410A. The even coefficients module 410A then obtainsthe even-modes coefficients by using equation:

$\begin{matrix}{{\begin{bmatrix}C_{mm} \\C_{{({m + 2})}m} \\\vdots \\C_{Nm}\end{bmatrix} = {\begin{bmatrix}U_{{mm}\; 1} & U_{{({m + 2})}m\; 1} & \ldots & U_{{Nm}\; 1} \\U_{{mm}\; 2} & U_{{({m + 2})}m\; 2} & \ldots & U_{{Nm}\; 2} \\\vdots & \vdots & \ddots & \vdots \\U_{{mm}\; X} & U_{{({m + 2})}m\; X} & \ldots & U_{{Nm}\; X}\end{bmatrix}^{- 1}\begin{bmatrix}\alpha_{m\; 1} \\\alpha_{m\; 2} \\\vdots \\\alpha_{m\; X}\end{bmatrix}}},} & \left( {{eq}.\mspace{14mu} 5} \right)\end{matrix}$

where

$\alpha_{mX} = {\frac{1}{N_{x}}{\sum\; {{P\left( {r,\theta,\phi,k} \right)}{E_{m}\left( {- \phi} \right)}}}}$

is the integral of mth order harmonics present on the Xth circular array310. P(r, θ, φ, k) are the outputs from the FFT modules 430B. Forexample, the outputs from the FFT modules 430B are measured soundpressure at frequency k from each microphone unit 340B. Equation 5 isthe inverse matrix of even-mode spherical harmonic coefficient, which isshown below in equation 23. For calculating the even-modes coefficients,m is an even number, and U_(mmX)=j_(n)(kr_(X))P^(nm)(cos θ) (eq. 6).

To determine the odd-modes spherical harmonics coefficients, the outputof the omnidirectional sensor 340A is digitized by the FFT module 430B.The differential module 440 then computes the differential between thedigitized outputs of the omnidirectional sensors 340A and 340B to obtainthe output of the first order sensor 350, as discussed in relation toFIG. 3A above. The output of the differential module 440 is transmittedto the odd coefficients module 420A.

The odd coefficients module 420A then obtains the odd-modes coefficientsby multiplying the input signal from the differential modules 440 withthe inverse matrix of odd-modes spherical harmonic coefficients, whichis shown below in equation 23.

Similar to obtaining the even-modes coefficients, the odd-modescoefficients are obtained using equation 5 hereinbefore, but P(r, θ, φ,k) are the outputs from the differential modules 440. For calculatingthe odd-modes coefficients, m is an odd integer, and

$\begin{matrix}{U_{mmX} = {{- \frac{\sin \left( \theta_{x} \right)}{{ikr}_{x}}}{j_{n}\left( {kr}_{X} \right)}{{P_{nm}^{\prime}\left( {\cos \mspace{11mu} \theta} \right)}.}}} & \left( {{eq}.\mspace{14mu} 7} \right)\end{matrix}$

Alternatively, the modules 410A and 420A may be implemented as a singlemodule that receives input from the FFT modules 430B and thedifferential modules 440. The merged matrix solution is:

$\begin{matrix}{\begin{bmatrix}C_{m\; m} \\C_{{({m + 1})}m} \\\vdots \\C_{Nm}\end{bmatrix} = {\begin{bmatrix}U_{m\; m\; 1} & U_{{({m + 1})}m\; 1} & \ldots & U_{{Nm}\; 1} \\U_{m\; m\; 2} & U_{{({m + 1})}m\; 2} & \ldots & U_{{Nm}\; 2} \\\cdots & \cdots & \ddots & \vdots \\U_{m\; {mX}} & U_{{({m + 1})}{mX}} & \ldots & U_{NmX}\end{bmatrix}^{- 1}\begin{bmatrix}\alpha_{m\; 1} \\\alpha_{m\; 2} \\\vdots \\\alpha_{mX}\end{bmatrix}}} & \left( {{eq}.\mspace{14mu} 8} \right)\end{matrix}$

As the 2D array 110A has circular arrays 310 with omnidirectionalsensors 340, equation 6 is used to calculate the U_(mmX).

Equation 8 is similar to equation 5, but the dimensions of the matricesof equation 8 are larger as the circular arrays 310 used here is the sumof both the omnidirectional and differential sensors 340B and 350.

FIG. 4B shows the implementation of the array 110B on the system 100.The output of each of the omnidirectional sensors 340 and the firstorder sensors 350 is transmitted to the computer system 120 which isthen processed by the array processing module 130B, as the arrayprocessing module 130B is executed by the processor 205.

The array processing module 130B includes FFT modules 432 and 434, evencoefficients module 410B, and odd coefficients module 420B. The FFTmodules 432 and 434 digitize the 3D wave signals acquired by theomnidirectional sensors 340 and the first order sensors 350,respectively. The even coefficients module 410B processes the digitized3D wave field signals, acquired by the omnidirectional sensors 340, tocalculate the even-mode spherical harmonic coefficients of the acquired3D wave field signals. The odd coefficients module 420B processes thedigitized 3D wave field signals, acquired by the first order sensors350, to calculate the even-mode spherical harmonic coefficients of theacquired 3D wave field signals.

To determine the even-modes spherical harmonics coefficients, theoutputs of the omnidirectional sensors 340 are transmitted to the FFTmodules 432 and the outputs of the FFT modules 432 are transmitted tothe even coefficients module 410B. The even coefficients module 410Bthen obtains the even-modes coefficients by using equation 5hereinbefore, but P(r, θ, φ, k) are the outputs from the FFT modules 432and 434, m is an even number, and equation 6 is used to calculate theU_(mmX).

To determine the odd-modes spherical harmonics coefficients, the outputsof the first order sensors 350 are digitized by the FFT modules 434 andtransmitted to the odd coefficients module 420B.

The odd coefficients module 420B then obtains the odd-modes coefficientsby using the equation: α_(diff)=α_(card)−U_(even)C_(even), whereα_(diff) is the α_(vector) of equation 5 for the differential sensor350, α_(card) is the α vector of equation 5 for the cardioid sensor 350,U_(even) is calculated using equation 6, and C_(even) is the even-modescoefficients obtained by the even coefficients module 410B.

The odd-modes coefficients are then calculated using equation 5, whereC_(odd)=[U_(odd)]⁻¹α_(diff), where U_(odd) obtained using equation 7,and α_(diff) is as defined in the previous paragraph.

Alternatively, the even coefficients module 410B and the oddcoefficients module 420B may be implemented as a single module, so thatthe operation can be performed using one matrix operation. The mergedmatrix solution is the same as equation 8, but U_(mmx) for this case is

$U_{mmX} = {{j_{n}\left( {kr}_{X} \right)}{\left( {{\beta \; {P_{n\; m}\left( {\cos \; \theta} \right)}} - \left( {1 - \alpha} \right) - {\frac{\sin \left( \theta_{x} \right)}{{ikr}_{x}}{P_{n\; m}^{\prime}\left( {\cos \; \theta} \right)}}} \right).}}$

β is a scaling factor, which is decided by the receiving pattern of thecardioid microphone 350.

The inverse matrix used by the modules 410A, 410B, 420A, and 420B isdependent on the position of the sensors 340, 350. Thus, the inversematrix is fixed for a specific configuration of 2D sensor array 110. Aplot of a reconstructed sound field and an actual sound field can befound in FIG. 4C. Plots (a) and (c) of FIG. 4C are the actual soundfield captured by the system of FIG. 1A (using the 2D sensor array 110A)at z=0 m and z=0.2 m plane respectively, whilst plots (b) and (d) arethe reconstructed wave field at these two planes respectively.

FIG. 5 shows a method 500 for the system 100 to acquire 3D wave signalsand construct a 3D wave field representation of the acquired 3D wavesignals. The method 500 commences with step 510 to acquire 3D wave fieldsignals using a 2D sensor array 110. The 2D sensor array 110 can beeither one of the example arrays 110A and 110B. The method 500 thenproceeds to step 520.

In step 520, the acquired 3D wave field signals are digitized. Theacquired 3D wave field signals are digitized by FFT modules 430 or 432and 434, depending on the configuration of the array 110 and asdescribed above in relation to FIGS. 4A and 4B. The method 500 thenproceeds to step 530.

In step 530, even coefficients of the spherical harmonics are computed.The digitized, acquired 3D wave field signals are transmitted to theeven coefficients module 410A or 410B. The transmission of the digitized3D wave field signals to the even coefficients module 410A or 410B is asdescribed in relation to FIGS. 4A and 4B above. The method 500 proceedsto step 540.

In step 540, odd coefficients of the spherical harmonics are computed.The digitized, acquired 3D wave field signals are transmitted to the oddcoefficients module 420A or 420B. The transmission of the digitized 3Dwave field signals to the odd coefficients module 420A or 420B is asdescribed in relation to FIGS. 4A and 4B above. The method 500 proceedsto step 550.

In step 550, a 3D wave field representation is constructed. Theconstruction of the 3D wave field representation (as defined inequation 1) occurs by using the computed even and odd coefficients. Themethod 500 concludes after step 550.

FIG. 6A shows an active noise cancellation system 600 having a 2D sensorarray 110, a computer system 120 with an array processing module 130 andan application module 140, and loudspeakers 640. The 2D sensor array 110and the array processing module 130 are as described above in relationto FIGS. 3A, 3B, 4A, and 4B. The computer system 120 is as describedabove in relation to FIGS. 2A and 2B.

The application module 140 includes an adaptive module 610, a filtermodule 620, and a signal generator module 630. The adaptive module 610receives the 3D wave field spherical harmonic coefficients (i.e., botheven- and odd-modes coefficients) from the array processing module 130.The adaptive module 610 then calculates a series of weightscorresponding to the noise received by array 110 and transmits theseries of weights to the filter module 620.

The filter module 620 receives the series of weights and adjusts filtercoefficients of the filter module 620. The filter coefficients are thentransmitted to the signal generator module 630.

The signal generator module 630 includes a reference signal matrix thathas been formed by multiplying the 3D wave field spherical harmoniccoefficients and channel (i.e., positional) information of loudspeakers640. The signal generator module 630 then multiplies the received filtercoefficients and the channel information to generate a series ofdiscrete loudspeaker driving signals for the loudspeakers 640. Thediscrete loudspeaker driving signals are complex numbers havingmagnitude and phase of a sinusoidal wave. The time-domain loudspeakerdriving signals are then generated by synthesizing the discreteloudspeaker driving signals and sent to the loudspeakers 640.

The loudspeakers' channel information may be calculated based ontheoretical models, or measured during an offline calibration process.The loudspeakers 640 play the driving signals generated by the signalgenerator module 630, and produce a sound field which is the phaseinversed version of the noise field, thus cancelling the noise andcreating a quiet zone. Such an active noise cancellation application canbe used as noise control within a cabin of vehicles (cars, airplanes),industrial noise reduction (factories), etc.

Conventional active noise cancelling techniques do not take into accountof the spatial sound information, therefore the performance of thesesystems are largely situational and limited. However, if carried out inthe wave domain (using sound field coefficients), the noise cancellationalgorithm can achieve a much greater resolution, both in frequencydomain and in spatial domain.

For example, the active noise cancellation can be used in a car. Forthis application, the microphone array 110 must be small enough to fitinto the ceiling of a car cabin, yet sufficiently large to provide goodresolution at all the target frequency bands. Since the majority of theambient noise power (e.g., road noise, wind noise and engine noise) lieswithin the range of 50-850 Hz, the microphone array 110 is designed tooperate within this frequency band.

Active noise cancellation requires high precision, but a small amount oferror could lead to significant drop in the performance of the system600. Therefore, the primary design goal for this application is tomaximize the precision of the system 600, while keeping the number ofmicrophones low.

For this application, the array 110A is better because omnidirectionalmicrophones 340 have a smaller profile compared to a cardioidmicrophone. The omnidirectional microphones 340 can also be mountedclose together without causing too much distortion to the acquired soundfield signals. Furthermore, the separate calculation of the even- andodd-modes coefficients of the array 110A minimizes the interferencebetween microphones 340 on different circular arrays 310, resulting inbetter precision.

The interior space of a car is often limited and irregular, which doesnot allow mounting of a spherical array inside the car. However, withthe 2D sensor array 110, it is possible to integrate the planar array110 to the ceiling of the car. With a maximum array radius 320N of 0.46metre, the array 110 can approximately cover the region where thepassengers' heads should be, while keeping the silhouette of the array110 relatively small. For a maximum frequency of 850 Hz and a radius320N of 0.46 m, the array 110 can receive sound field harmonics up toorder

$\begin{matrix}{N = {\left\lceil \frac{ekr}{2} \right\rceil = 10}} & \left( {{eq}.\mspace{14mu} 9} \right)\end{matrix}$

which means that at least 2N+1=27 microphone pairs (350) should beplaced on the outermost circular array 310E. The circular arrays 310 areplaced non-uniformly with more circular arrays 310 being closer to theouter circular array 310E, to maximize the detection accuracy of higherorder harmonics. For this example, the radius 320 are set to 0.46 m,0.38 m, 0.30 m and 0.20 m. The amount of microphones 350 on eachcircular array 310A, 310B, 310C, 310D, 310E is determined to be 21, 19,15 and 11, respectively.

The performance of the array 110 for active noise cancellation isevaluated using simulations. The simulations test the response of thearray 110 against a single point source of frequency 150-1150 Hzimpinging from 1.6 metres away from the centre of the array at θ=π/4.The derived sound field coefficients are used to reconstruct the 3D wavefield by comparing the reconstructed 3D wave field and the originalsound field.

FIG. 6B shows the performance of the system 600 for differentfrequencies and impinging angles. It can be seen from FIG. 6B that erroris relatively small below 850 Hz, which is the design maximum frequencyof the array 110. Beyond 850 Hz, the error increases rapidly as theorder of spherical harmonics increases beyond the design frequency ofthe array 110, as determined by the method 300.

According to FIG. 6B, the reproduction accuracy does not changesignificantly with the impinging angle of the plane wave. At 850 Hz, thearray 110 is able to capture waves impinging from any direction withless than 4% of error.

FIG. 7 shows a beamforming system 700 having a 2D sensor array 110, acomputer system 120 with an array processing module 130 and anapplication module 140, and loudspeakers/antennas 740. The 2D sensorarray 110 and the array processing module 130 are as described above inrelation to FIGS. 3A, 3B, 4A, and 4B. The computer system 120 is asdescribed above in relation to FIGS. 2A and 2B.

The application module 140 includes a weighting module 710, a wavesynthesizer module 720, and an adaptive module 730. The weighting module710 receives the 3D wave field coefficients from the array processingmodule 130 and multiplies each of the received coefficients with aweighting factor. The weighted coefficients are then transmitted to thewave synthesizer module 720, which uses the weighted coefficients tosynthesize a new time domain signal. The synthesized time domain signalis the output of the beamformer.

The adaptive module 730 optimizes the directivity of the system 700. Theadaptive module 730 receives the beamforming output of the wavesynthesizer module 720 and compares the beamforming output with a targetbeamforming direction. The adaptive module 730 then updates theweighting factors depending on the comparison and transmits the updatedweighting factors to the weighting module 710. Typical applications ofthe beamforming system 700 are RF antenna arrays, directional audiorecording (such as for conference recording), etc.

FIG. 8 shows a source localization/direction of arrival estimationsystem 800 having a 2D sensor array 110, a computer system 120 with anarray processing module 130 and an application module 140. The 2D sensorarray 110 and the array processing module 130 are as described above inrelation to FIGS. 3A, 3B, 4A, and 4B. The computer system 120 is asdescribed above in relation to FIGS. 2A and 2B.

The application module 140 includes a correlation matrix module 810 anda direction of arrival module 820. The correlation matrix module 810calculates a correlation matrix from the constructed 3D wave fieldrepresentation and transmits the correlation matrix to the direction ofarrival module 820. The direction of arrival module 820 includes adirection of arrival algorithm (e.g., MUSIC) which uses the correlationmatrix to output a two-dimensional plot corresponding to a possibledirection of an impinging wave. Applications that can use the directionof arrival system 800 are tracking, sonar/radar scanning, etc.

FIG. 9 shows a 3D sound field recording system 900 having a 2D sensorarray 110, a computer system 120 with an array processing module 130 andan application module 140, and loudspeakers 940. The 2D sensor array 110and the array processing module 130 are as described above in relationto FIGS. 3A, 3B, 4A, and 4B. The computer system 120 is as describedabove in relation to FIGS. 2A and 2B.

The application module 140 includes a wave synthesizer module 910 forreceiving the constructed 3D wave field representation and generating aseries of loudspeakers driving signals. The wave synthesizer module 910requires prior knowledge of the position and channel information of theloudspeakers 940 to generate the loudspeakers driving signals. The wavesynthesizer module 910 then transmits the generated loudspeakers drivingsignals to the loudspeakers 940. However, unlike the system 600, anadaptive module is not required as the accuracy in reconstructing theacquired 3D wave field signals is not required to be as precise as thesystem 600.

The system 900 allows complete recording and synthesis of an acousticscene. The system 900 can be used for applications such asteleconferencing, audio recording for performances/movies, etc.

Mathematical Proof of Obtaining Odd-Modes Spherical HarmonicsCoefficient on the x-y Plane

As mentioned in the Background section above, a 3D wave field sphericalharmonics decomposition requires sensors located in 3D (i.e., xyzplane). That is, sensors placed only on the x-y plane (i.e., 2D) wouldnot be able to provide the odd-modes 1010 of the spherical harmonics, asthe odd-modes 1010 on x-y plane would be zero. However, it was observedthat FIG. 10 shows that the changing rate of P_(nm) with respect to θ

$\left( {{i.e.},\frac{{dP}_{n}^{m}\left( {\cos (\theta)} \right)}{d\; \theta}} \right)$

is identically opposite for even- and odd-modes. That is, the even-modes1020 of

$\frac{{dP}_{n}^{m}\left( {\cos (\theta)} \right)}{d\; \theta}$

is equal to zero when

${\theta = \frac{\pi}{2}},$

and the odd modes 1010 are non-zero at

$\theta = {\frac{\pi}{2}.}$

The observation is proven by a property of the Associated LegendreFunction:

$\begin{matrix}{{\left( {x^{2} - 1} \right)\frac{{dP}_{n\; m}(x)}{dx}} = {{{nxP}_{n\; m}(x)} - {\left( {m + n} \right){P_{{n - 1},m}(x)}}}} & \left( {{eq}.\mspace{14mu} 10} \right)\end{matrix}$

When

${x = {{\cos \left( \frac{\pi}{2} \right)} = 0}},$

equation 10 becomes P′_(nm)(0)=(m+n)P_(n-1,m)(0).

This proves the observed relationship between

${P_{n\; m}^{\prime}\left( {\cos \left( \frac{\pi}{2} \right)} \right)}\mspace{14mu} {and}\mspace{14mu} {{P_{{n - 1},m}\left( {\cos \left( \frac{\pi}{2} \right)} \right)}.}$

Therefore, P_(n-1,m)(0)=0=P′_(n,m)(0). Otherwise, both P_(n-1,m)(0) andP′_(nm)(0) would be non-zero. As the odd-number

$P_{n{m}}\left( {\cos \left( \frac{\pi}{2} \right)} \right)$

is zero and the even-number

$P_{n{m}}\left( {\cos \left( \frac{\pi}{2} \right)} \right)$

is non-zero, it can be seen that the even-number

$P_{n{m}}^{\prime}\left( {\cos \left( \frac{\pi}{2} \right)} \right)$

would be zero, while the odd-number

$P_{n{m}}^{\prime}\left( {\cos \left( \frac{\pi}{2} \right)} \right)$

is non-zero.

Therefore, although the odd-number

$P_{n{m}}\left( {\cos \left( \frac{\pi}{2} \right)} \right)$

is not observable on the x-y plane, the odd-number

$P_{n{m}}^{\prime}\left( {\cos \left( \frac{\pi}{2} \right)} \right)$

is observable. This means that the odd-modes can be determined.

The above mathematical relations therefore prove that the odd-modes of3D wave field spherical harmonics decomposition can be obtained on thex-y plane.

First Order Sensor for Acquiring the Odd-Mode Spherical HarmonicsCoefficient on the x-y Plane

In the spherical coordinate system, the particle velocity at a point (r,θ, φ) can be represented in the frequency domain as the gradient ofequation 1:

$\begin{matrix}{{V\left( {r,\theta,\varphi,k} \right)} = {\frac{1}{i\; \varrho_{0}{ck}}{\nabla{P\left( {r,\theta,\varphi,k} \right)}}}} & \left( {{eq}.\mspace{14mu} 11} \right)\end{matrix}$

For simplicity, the sensor is assumed to measure particle velocity inthe θ direction. Equation 11 can then be modified to be:

$\begin{matrix}{{V\left( {r,\theta,\varphi,k} \right)} = {\frac{1}{i\; \varrho_{0}{ck}}\frac{\partial{P\left( {r,\theta,\varphi,k} \right)}}{\partial\theta}}} & \left( {{eq}.\mspace{14mu} 12} \right)\end{matrix}$

Substituting equation 1 into equation 12 yields:

$\begin{matrix}{{V\left( {r,\theta,\varphi,k} \right)} = {\frac{1}{i\; \varrho_{0}{ck}}\frac{\partial{\sum_{n = 0}^{\infty}{\sum_{m = {- n}}^{n}\; {C_{nm}{j_{n}({kr})}_{n{m}}{E_{m}(\varphi)}}}}}{\partial\theta}}} & \left( {{eq}.\mspace{14mu} 13} \right)\end{matrix}$

Taking the partial derivative of P_(n) ^(m)(cos(θ)) yields:

$\begin{matrix}{{V\left( {r,\theta,\varphi,k} \right)} = {\frac{{- \sin}\mspace{11mu} \theta}{i\; \varrho_{0}{ck}}{\sum\limits_{n = 0}^{\infty}{\sum\limits_{m = {- n}}^{n}\; {C_{nm}{j_{n}({kr})}{_{n{m}}^{\prime}\left( {\cos \mspace{11mu} \theta} \right)}{E_{m}(\varphi)}}}}}} & \left( {{eq}.\mspace{14mu} 14} \right)\end{matrix}$

Since the coefficients, C_(nm), are treated as unknown parameters inequation 14, the derived representation should be suitable to representany types of wave field.

For a plane wave impinging from a direction (θ, φ), the wave field canbe described as P(r, θ, φ, k)=e^(ikr(sin θ sin θ cos (φ−θ+cos θ cos θ)),which when substituted into equation 12 yields:

$\begin{matrix}{V = {\frac{1}{i\; \varrho_{0}c}{i\left( {{\sin \mspace{11mu} \theta \mspace{11mu} \sin \mspace{11mu} \vartheta \mspace{11mu} \cos \mspace{11mu} \left( {\varphi - \phi} \right)} + {\cos \mspace{11mu} \theta \mspace{11mu} \cos \mspace{11mu} \vartheta}} \right)}P}} & \left( {{eq}.\mspace{14mu} 15} \right)\end{matrix}$

Combining equations 14 and 15 gives the wave field captured by a firstorder sensor as:

$\begin{matrix}{{\left( {{\cos \mspace{11mu} \theta \mspace{11mu} \sin \mspace{11mu} \vartheta \mspace{11mu} \cos \mspace{11mu} \left( {\varphi - \phi} \right)} - {\sin \mspace{11mu} \theta \mspace{11mu} \cos \mspace{11mu} \vartheta}} \right){P\left( {r,\theta,\varphi,k} \right)}} = {{- \frac{\sin \mspace{11mu} \theta}{ikr}}{\sum\limits_{n = 0}^{\infty}{\sum\limits_{m = {- n}}^{n}\; {C_{nm}{j_{n}({kr})}{_{n{m}}^{\prime}\left( {\cos \mspace{11mu} \theta} \right)}{E_{m}(\varphi)}}}}}} & \left( {{eq}.\mspace{14mu} 16} \right)\end{matrix}$

A part of the left hand side of equation 16 (i.e., (cos θ sin θcos)φ−φ)−sin θ cos θ).) is equal to the pattern of a first order sensorwhen normally placed to the radial direction at θ=π/2. That is, thefirst order sensor is pointed at

$\left( {{\theta + \frac{2}{\pi}},\varphi} \right).$

The remaining term on the left hand side (i.e., P(r, θ, φ, k)) is theactual wave field at the position of the sensor. Multiplication of thetwo left-hand side terms gives the wave field acquired by the firstorder sensor.

On the right hand side of equation 16, the Associated Legendre functionis replaced by the first order derivative of the function, which changesthe zero points of the term. As discussed above, the odd modes of P_(n)^(m)(x) are non-zero when x=0. This new representation therefore allowsmeasurement of odd-mode wave field coefficients on the

$\theta = \frac{\pi}{2}$

plane.

Thus, according to paragraphs [00122] and [00123] above, a first ordersensor is capable of obtaining the odd mode coefficients of thespherical harmonics when the first order sensor is placed vertically(i.e., perpendicular to the x-y plane).

One example implementation of a first order sensor is twoomnidirectional sensors placed in close proximity, as discussed inrelation to FIG. 3A. Assuming that the two omnidirectional sensors areplace in a spherical coordinate with location M₁=(r,0, 0) and M₂=(r, π,0), the output signal is defined as

S=S ₁ −S ₂  (eq. 17)

where S₁ and S₂ are signal outputs of the two omnidirectional sensorslocated at M₁ and M₂, respectively. For a far field source located at aposition (L, θ, φ)(L>>r), the two omnidirectional sensors has areception pattern represented as:

G=cos θ  (eq. 18)

Although the magnitudes of the lobes are symmetric, the lobe in the

$\theta > \frac{\pi}{2}$

region has a negative gain, meaning the phase of the signal is invertedif the signal source comes from this direction. The magnitudes of thelobes are critical as the direction of the sensor pair must match withequation 16. Otherwise, the calculated wave field would be inverted.

In another example, cardioid sensors can be used as first order sensors.Typically, the cardioid pattern of cardioid sensors is realized bytaking the sum of two captured signals and applying a delay to one ofthe captured signals. The applied delay determines the gain pattern ofthe cardioid sensor. For audio microphones, the delay is realizedthrough utilizing materials that reduce the speed of sound wave.

In one example, the cardioid sensor can be implemented using twoomnidirectional sensors and applying a delay to an output of one of theomnidirectional sensors. It should be noted that a cardioid sensor doesnot offer superiority over the two omnidirectional sensorsimplementation.

A cardioid pattern can also be obtained as the weighted sum of anomnidirectional pattern and a differential pattern, the gain can thus bewritten as:

G=(β+(1−β)cos(θ))  (eq. 19)

where β is the weighting factor. A standard cardioid pattern is realizedwhen β=0.5, meaning the two components have the same weight. Thus, thewave field captured by a cardioid sensor can be represented as theweighted sum of an omnidirectional sensor and a differential sensor.Combining equations 1 and 14 yields the wave field acquired by acardioid sensor:

$\begin{matrix}{{\left( {\beta + {\left( {1 - \beta} \right)\left( {{\cos \mspace{11mu} \theta \mspace{11mu} \sin \mspace{11mu} \vartheta \mspace{11mu} \cos \mspace{11mu} \left( {\varphi - \phi} \right)} - {\sin \mspace{11mu} \theta \mspace{11mu} \cos \mspace{11mu} \vartheta}} \right)}} \right)P} = {\sum\limits_{n = 0}^{\infty}{\sum\limits_{m = {- n}}^{n}\; {C_{nm}{j_{n}({kr})}\left( {{\beta \; _{n{m}}} - {\left( {1 - \beta} \right)\frac{\sin \mspace{11mu} \theta}{ikr}_{n{m}}^{\prime}}} \right){E_{m}(\varphi)}}}}} & \left( {{eq}.\mspace{14mu} 20} \right)\end{matrix}$

Similar to equation 16, the left hand side of equation 20 represent thereception pattern of a cardioid sensor multiplied by the actual wavefield at the position of the sensor. The right hand side of equation 20is the weighted combination of the wave field decomposition of the tworeception patterns. For a given cardioid sensor (e.g., a unidirectionalmicrophone), the left hand side of equation 20 is fixed. However, if theexact beam pattern is known, the weighting coefficient α can be adjustedto match the beam pattern.

Array Configuration

The total wave field at location

$\left( {r,\frac{\pi}{2},\varphi} \right)$

over a circular array 310 of the array 110 can be obtained by theequation:

$\begin{matrix}{{P\left( {r,\theta,\varphi,k} \right)} = {\sum\limits_{n = 0}^{\infty}{\sum\limits_{m = {- n}}^{n}\; {C_{nm}{j_{n}({kr})}{_{n{m}}(0)}{E_{m}(\varphi)}}}}} & \left( {{eq}.\mspace{14mu} 21} \right)\end{matrix}$

Multiplying both sides of equation 21 by E_(m)(−φ) and integrating withrespect to Φ over [0, 2π) then yields the total wave field over oneconcentric circular array (e.g., 310A):

$\begin{matrix}{{\int_{0}^{2\; \pi}{{{PE}_{m}\left( {- \varphi} \right)}\ d\; \varphi}} = {\sum\limits_{n = {m}}^{N}\; {C_{nm}{j_{n}({kr})}{_{n{m}}(0)}}}} & \left( {{eq}.\mspace{14mu} 22} \right)\end{matrix}$

The total wave field over a plurality of concentric circular arrays 310is found through solving the matrix equation:

$\begin{matrix}{\begin{bmatrix}\alpha_{m\; 1} \\\alpha_{m\; 2} \\\vdots \\\alpha_{m\; X}\end{bmatrix} = {\begin{bmatrix}U_{{mm}\; 1} & U_{{({m + 1})}m\; 1} & \ldots & U_{N\; m\; 1} \\U_{{mm}\; 2} & U_{{({m + 1})}m\; 2} & \ldots & U_{N\; m\; 2} \\\vdots & \vdots & \ddots & \vdots \\U_{{mm}X} & U_{{({m + 1})}m\; X} & \ldots & U_{N\; m\; X}\end{bmatrix}*\begin{bmatrix}C_{mm} \\C_{{({m + 1})}m} \\\vdots \\C_{Nm}\end{bmatrix}}} & \left( {{eq}.\mspace{14mu} 23} \right)\end{matrix}$

Where α_(mX)=∫₀ ^(2π)P(r, θ, φ, k)E_(m)(−φ)dφ is the integral of mthorder harmonics present on a circular array (e.g., 310A).U_(NmX)=j_(N)(kr_(X))P_(N|m|)(0) is the harmonic associated with eachcoefficient.

Since the term P_(N|m|)(0)=0 for odd-modes, only even mode coefficientscan be derived from the above equation.

One example in deriving odd mode coefficients is to use first ordersensors placed on a circular array (e.g., 310A) and pointedperpendicular to the

$\theta = \frac{\pi}{2}$

plane. Equation 16 is then integrated with respect to φ over [0, 2π) toyield:

$\begin{matrix}{{\int_{0}^{2\pi}{P_{\theta}{E_{m}\left( {- \varphi} \right)}d\; \varphi}} = {{- \frac{\sin \; \theta}{ikr}}{\sum\limits_{n = {m}}^{N}\; {C_{nm}{j_{n}({kr})}\left( P^{\prime} \right)_{n{m}}\left( {\cos \; \theta} \right)}}}} & \left( {{eq}.\mspace{14mu} 24} \right)\end{matrix}$

where

${P_{\theta}\left( {r,\frac{\pi}{2},\varphi,k} \right)} = {\left( {{\cos \frac{\pi}{2}\sin \; {{\theta cos}\left( {\varphi - \phi} \right)}} - {\sin \frac{\pi}{2}\cos \; \theta}} \right){P\left( {r,\frac{\pi}{2},\varphi,k} \right)}}$

is the wave field received by a first order sensor at (r, π/2, Φ). Thecoefficients can then be calculated by solving the matrix inversionpresented in equation 23, with

$U_{NmX} = {{- \frac{\sin \left( \theta_{X} \right)}{{ikr}_{X}}}{j_{N}\left( {kr}_{X} \right)}{{P_{N{m}}^{\prime}\left( {\cos \; \theta_{X}} \right)}.}}$

Due to the term

_(n|m|)(cos θ)=0 when

${\theta = \frac{\pi}{2}},$

the matrix solution only contains the odd mode coefficients if the firstorder sensors are placed on a circular array (e.g., 310A).

The full 3D wave field spherical harmonics decomposition can thereforebe retrieved by combining the retrieved even and odd mode coefficients.

Equations 22 and 24 are for an ideal situation where the sensor iscontinuous over a circular array (e.g., 310A). In a real-worldimplementation, however, there are only a discrete number of sensorsplaced on a circular array (e.g., 310A). Thus, the discrete forms ofequations 22 and 24 are:

$\begin{matrix}{{{\frac{2\pi}{N_{X}}{\sum\; {{P\left( {r_{X},\theta_{X},\varphi,k} \right)}{E_{m}(\varphi)}}}} = {\sum\limits_{n = {m}}^{N}\; {C_{nm}{j_{n}({kr})}{P_{n{m}}(0)}}}}{and}} & \left( {{eq}.\mspace{14mu} 25} \right) \\{{\frac{2\pi}{N_{X}}{\sum\; {{P_{\theta}\left( {r_{X},\theta_{X},\varphi,k} \right)}{E_{m}(\varphi)}}}} = {{- \frac{\sin \; \theta}{ikr}}{\sum\limits_{n = {m}}^{N}\; {C_{nm}{j_{n}({kr})}{P_{n{m}}^{\prime}\left( {\cos \; \theta} \right)}}}}} & \left( {{eq}.\mspace{14mu} 26} \right)\end{matrix}$

Due to the spatial sampling of the acquired 3D wave field signals,sensors (340 or 350) placed on a circular array (e.g., 310A) observe alimited number of spherical harmonic orders. The relationship betweenthe number of sensors N_(X) and the maximum observable sphericalharmonic order is given by N_(X)=2N+1. However, the nature of thespherical Bessel function means a limited number of orders is observablewithin a circular array (e.g., 310A). The maximum number of observableharmonic orders is given by the equation:

$\begin{matrix}{N \leq \left\lceil \frac{ekr}{2} \right\rceil} & \left( {{eq}.\mspace{14mu} 27} \right)\end{matrix}$

where k is the wave number. The exact amount of sensor N_(X) to be usedfor each circular array (e.g., 310A) depends on the size of the circulararray (e.g., 310A) as well as the wave number k of the target wavefield.

As the number of spherical harmonic orders obtainable by a circulararray (e.g., 310A) is limited, aliasing of higher order harmonics (whichare not detected by the circular array 310A) into the lower orderharmonics occurs. The aliasing creates an error which depends on thepresence of undetected high order harmonics. However, equation 27 givessufficient number of sensors in each circular array 310 to provide highprecision for most applications.

If an application however does not demand a high accuracy, a lowerprecision can be used to design a circular array (e.g, 310A). The lowerprecision assumes the observable harmonic order at a radius r is N≦┌kR┐,which means less sensors would be used for a circular array (e.g.,310A).

Since the quantity of sensors (340 or 350) on a circular array (e.g.,310A) is linked to the wave number k, which can be translated into thewavelength λ, the number of sensors (340 or 350) can be derived from thetarget frequency of the application, given by equation:

$\begin{matrix}{N_{X} = {{{2N} + 1} = {{{2\left\lceil \frac{ekr}{2} \right\rceil} + 1} = {{{2\left\lceil \frac{{e\pi}\; r}{\lambda} \right\rceil} + 1} = {{2\left\lceil \frac{e\; \pi \; {fR}}{c} \right\rceil} + 1}}}}} & \left( {{eq}.\mspace{14mu} 28} \right)\end{matrix}$

where c is the speed of the wave. For audio applications, c=340 m/s; forRF applications, c=300,000,000 m/s. For example, a circular array (e.g.,310A) of 0.2 m radius, designed to receive 900 MHz RF signal would have13 sensors; while a circular array (e.g., 310A) of 0.4 m radius,designed for audio signals up to 1500 Hz would need 33 sensors (i.e.,microphones).

The total number of sensors (340 or 350) is doubled for array 110A, dueto the use of two omnidirectional sensors 340 for each sensor 350. Itshould be noted that the distance between the two omnidirectionalsensors 340 in array 110A is small compared to the array radius 320N, toapproximate P_(n) ^(m)(cos θ) in equation 24.

For arrays 110A and 110B, the even- and odd-modes coefficients can becalculated together using a single pseudo-inverse operation, instead ofhaving to process either of them individually. The calculation is donein the same manner as equation 23, with α_(mX) being the sum of mth moderesponse of a circular array (e.g., 310A). U_(NmX) is the wave fieldexpression for a circular array (e.g., 310A), and it has to exactlymatch with α_(mX) on the left hand side of the equation. In other words,U_(Nmx) should take one of the following expressions, based on thesensor (340 or 350) used on a circular array (e.g., 310A):

$\begin{matrix}{U_{NmX} = \left\{ \begin{matrix}{{j_{N}\left( {kr}_{X} \right)}P_{N{m}}} & {omnidirectional} \\{{- \frac{\sin \left( \theta_{X} \right)}{{ikr}_{X}}}{j_{N}\left( {kr}_{X} \right)}P_{N{m}}^{\prime}} & {differential} \\{{j_{n}({kr})}\left( {{\alpha \; P_{n{m}}} - {\left( {1 - \alpha} \right)\frac{\sin \; \theta}{ikr}P_{n{m}}^{\prime}}} \right)} & {cardioid}\end{matrix} \right.} & \left( {{eq}.\mspace{14mu} 29} \right)\end{matrix}$

The 2D array's ability to detect vertical components (i.e., odd-modecoefficients) of the 3D wave field spherical harmonic decomposition isdue to the combined use of zeroth (340) and first order sensors (350). Aplanar array of omnidirectional sensors (340) cannot distinguish betweenwaves that come from either sides of the array 400. As a result, onlythe even modes of the wave field components, which are symmetric overthe sensor plane, can be picked up by a planar array. The odd modescorrespond to wave field components that are non-symmetric over theplane, these modes are essential to represent a wave impinging from anoff-plane direction. The combined use of omnidirectional sensors andvertically placed first order sensors allow the array to distinguishbetween waves impinging from either sides of the array plane, whichmeans that the non-symmetric portion of the wave field is now visible tothe hybrid sensor array.

As mentioned in paragraph [0008] above, the spherical array works at anarrow frequency band. The 2D array 110 however operates on a broadfrequency band, since the array 110 is distributed uniformly. Some ofthe circular arrays 310 would always be able to receive an active mode,at certain frequencies. However, the highest order active modes arereceived by the one or two largest circular arrays of radius 320N or320N−1. Therefore the radius 320 of the circular arrays 310 need to bedesigned cautiously, otherwise the captured wave field coefficients maybe noisy. Overall, the 2D array 110 has the capability of analysingbroad band 3D wave field signals. Such a broad band capability makes the2D array 110 suitable for acoustic applications, as acoustic waves suchas human voice and engine noise have a wide frequency band, typicallyfrom 100 Hz to a few thousand Hertz.

Error Analysis Spatial Sampling

In most sensor array configurations that are based on discrete sensors,spatial sampling is a major and fundamental cause of error. Typically,in order to avoid spatial sampling, the distance between each twosensors must be less than half of the wavelength of the target signal

$\left( {{i.e.},{d \leq \frac{\lambda}{2}}} \right),$

where d is the distance between two sensors and λ is the wavelength ofthe highest frequency of the signal.

For the 2D array 110 described above, however, the error due to spatialsampling is represented by the equation:

$\begin{matrix}{{\Delta \; E} = {{\int_{0}^{2\pi}{{P\left( {r,\theta,\varphi,k} \right)}{E_{m}\left( {- \varphi} \right)}d\; \varphi}} - {\frac{2\pi}{N_{X}}{\sum\; {{P\left( {r_{X},\theta,\varphi,k} \right)}{E_{m}\left( {- \varphi} \right)}}}}}} & \left( {{eq}.\mspace{14mu} 30} \right)\end{matrix}$

The spatial sampling of the array 110 limits the maximum observablespatial frequency, which is represented as the order of the sphericalharmonics. The truncation error due to the limitation is represented bythe equation:

$\begin{matrix}{\begin{matrix}{{\Delta \; E} = {{\sum\limits_{n = {m}}^{\infty}\; {C_{nm}{j_{n}({kr})}P_{n{m}}}} -}} \\{= {\sum\limits_{n = {N + 1}}^{\infty}{C_{nm}{j_{n}({kr})}P_{n{m}}}}}\end{matrix}{\sum\limits_{n = {m}}^{N}\; {C_{nm}{j_{n}({kr})}P_{n{m}}}}} & \left( {{eq}.\mspace{14mu} 31} \right)\end{matrix}$

The power ΔE depends on the spherical Bessel function, which value is afunction of the wave number k and the radius 320 (i.e., r). For a givenorder of the spherical Bessel function j_(N)(kr), if kr is sufficientlysmall, then the value of j_(N)(kr) becomes close to zero. On the otherhand, for a fixed kr, the active spherical harmonics is limited to N.

The maximum number of active spherical harmonics order is given by

$N = {\left\lceil \frac{ekr}{2} \right\rceil.}$

The minimum amount of sensor to be used in the array is therefore 2N+1.However, the array's accuracy can be improved through the use ofadditional sensors in the array.

In equation 25, the wave field coefficients are calculated throughsolving the matrix equation r=. UC. Any aliasing error on a impacts onal the sound field coefficient C_(n) ^(m). The error percentage is givenby:

$\begin{matrix}{E_{m} = {\frac{\Delta \; E_{mX}}{\alpha_{mX}} = \frac{\sum\limits_{n = {N + 1}}^{\infty}{C_{nm}{j_{n}({kr})}P_{n{m}}}}{\sum\limits_{n = {m}}^{\infty}{C_{nm}{j_{n}({kr})}P_{n{m}}}}}} & \left( {{eq}.\mspace{14mu} 32} \right)\end{matrix}$

If a circular array (e.g., 310A) has insufficient amount of sensors (340or 350), then all the α_(mX) derived from the array's output would beincorrect. Since the algorithm uses a matrix inversion operation tosolve the coefficients, which follows the least mean square (LMS) fit,the errors in a single α_(mX) are evenly distributed to all of thecalculated wave field coefficients of mode m. The error at eachresulting coefficient C_(nm) is given by:

$\begin{matrix}{E_{nm} = \frac{E_{m}}{n - m + 1}} & \left( {{eq}.\mspace{14mu} 33} \right)\end{matrix}$

Equation 33 indicates that individual coefficients of lower modes tendto suffer less from the aliasing error, since there are morecoefficients to share the error. However, the total error power of eachmode remains the same across all available modes.

The error percentage can be minimized as long as a sufficient amount ofsensors are used for the spatial sampling. It should be noted that theerror percentage only reflects the average expected error of calculatedcoefficients due to spatial aliasing. Another source of error due toinsufficient sampling is that a certain amount of active high orderharmonics are missed out (these harmonics caused aliasing error to thelower order coefficients), but this type of error is only visible duringwave field reproduction and the individual wave field coefficients arenot directly affected by this phenomenon.

As the array has multiple concentric circular arrays 310, each circulararray (e.g., 310A) observes a different maximum order of sphericalharmonics. For example, all the circular arrays 310 observe the zerothorder harmonic, but the highest order harmonics is observable by thelargest circular array 310N of radius 320N. Thus, calculation of thelower order modes is more accurate, as the lower order modes are sampledby most (if not all) of the circular arrays 310. The higher ordercoefficients however are not as precisely calculated, as the highestorder modes are only observed by one or two circular arrays (e.g, 310Nand 310N−1) with the largest radius (e.g., 320N and 320N−1,respectively). This results in the central part of the wave field beingcalculated more accurately than the outer area. The reconstructed 3Dwave field thus gradually loses precision at greater elevations, sinceonly the high order harmonics contribute to these regions. In order tocompensate for such an error, more circular arrays 310 with largerradius 320 be placed in the system 100, rather than using an uniformlydistributed circular arrays 310.

INDUSTRIAL APPLICABILITY

The arrangements described are applicable to the signal processingfield.

The foregoing describes only some embodiments of the present invention,and modifications and/or changes can be made thereto without departingfrom the scope and spirit of the invention, the embodiments beingillustrative and not restrictive.

In the context of this specification, the word “comprising” means“including principally but not necessarily solely” or “having” or“including”, and not “consisting only of”. Variations of the word“comprising”, such as “comprise” and “comprises” have correspondinglyvaried meanings.

1. A method for constructing a three-dimensional (3D) wave fieldrepresentation of a 3D wave field using a two-dimensional (2D) sensorarray, said method comprising: acquiring 3D wave field signals using a2D array of sensors, said 2D array of sensors comprising omnidirectionalsensors and first order sensors arranged in a 2D plane; digitizing saidacquired 3D wave field signals; computing even coefficients of sphericalharmonics dependent upon said digitized 3D wave field signals acquiredby said omnidirectional sensors; computing odd coefficients of saidspherical harmonics dependent upon said digitized 3D wave field signalsacquired by said first order sensors; and constructing a 3D wave fieldrepresentation dependent upon said computed even and odd coefficientsfor said acquired 3D wave field signals.
 2. The method according toclaim 1, wherein the omnidirectional sensors and the first order sensorsare placed on a plurality of concentric circular arrays.
 3. The methodaccording to claim 2, wherein the first order sensors comprise twoomnidirectional sensors and each of outputs of the first order sensorsis dependent on a differential between outputs of the twoomnidirectional sensors.
 4. The method according to claim 3, wherein thecomputing of the odd coefficients of said spherical harmonics depends onthe outputs of the first order sensors, and the computing of the evencoefficients of said spherical harmonics depends on the output of one ofthe two omnidirectional sensors.
 5. The method according to claim 2,wherein the concentric circular arrays with the omnidirectional sensorsare alternated with the concentric circular arrays with the first ordersensors, and wherein the first order sensors include cardioid sensors.6. The method according to claim 5, wherein the computing of the oddcoefficients of said spherical harmonics depends on outputs of the firstorder sensors, and the computing of the even coefficients of saidspherical harmonics depends on outputs of the omnidirectional sensors.7. The method according to claim 1, wherein the constructed 3D wavefield representation is used for any one of the following applications:active noise cancellation; beamforming; direction of arrival estimation;and sound recording or reproduction.
 8. An apparatus configured toconstruct a three-dimensional (3D) wave field representation of a 3Dwave field using a two-dimensional (2D) sensor array, the apparatuscomprising: a two-dimensional (2D) array of sensors configured toreceive the 3D wave field signals, said 2D array of sensors comprisingomnidirectional sensors and first order sensors arranged in a 2D plane;and a processor configured to: receive the acquired 3D wave fieldsignals; digitize said acquired 3D wave field signals; compute evencoefficients of spherical harmonics dependent upon said digitized 3Dwave field signals acquired by said omnidirectional sensors; compute oddcoefficients of said spherical harmonics dependent upon said digitized3D wave field signals acquired by said first order sensors; andconstruct a 3D wave field representation dependent upon said computedeven and odd coefficients for said acquired 3D wave field signals. 9.The apparatus according to claim 8, wherein the omnidirectional sensorsand the first order sensors are placed on a plurality of concentriccircular arrays.
 10. The apparatus according to claim 9, wherein thefirst order sensors comprise two omnidirectional sensors and each ofoutputs of the first order sensors is dependent on a differentialbetween outputs of the two omnidirectional sensors.
 11. The apparatusaccording to claim 10, wherein the processor computes the oddcoefficients of said spherical harmonics depending on the outputs of thefirst order sensors, and the processor computes the even coefficients ofsaid spherical harmonics depending on the output of one of the twoomnidirectional sensors.
 12. The apparatus according to claim 9, whereinthe concentric circular arrays with the omnidirectional sensors arealternated with the concentric circular arrays with the first ordersensors, and wherein the first order sensors include cardioid sensors.13. The apparatus according to claim 12, wherein the processor computesthe odd coefficients of said spherical harmonics depending on outputs ofthe first order sensors, and the computing of the even coefficients ofsaid spherical harmonics depending on outputs of the omnidirectionalsensors.
 14. The apparatus according to claim 8, wherein the constructed3D wave field representation is used for any one of the followingapplications: active noise cancellation; beamforming; direction ofarrival estimation; and sound recording or reproduction.
 15. Theapparatus according to claim 8, wherein the processor comprises adigital signal processor.
 16. The apparatus according to claim 8,wherein the processor comprises a field-programmable gate array.
 17. Acomputer readable storage medium storing computer instructions forconstructing a three-dimensional (3D) wave field representation of a 3Dwave field using a two-dimensional (2D) sensor array, said instructionconfigured to cause a computer apparatus to perform the steps of:acquiring 3D wave field signals using a 2D array of sensors, said 2Darray of sensors comprising omnidirectional sensors and first ordersensors arranged in a 2D plane; digitizing said acquired 3D wave fieldsignals; computing even coefficients of spherical harmonics dependentupon said digitized 3D wave field signals acquired by saidomnidirectional sensors; computing odd coefficients of said sphericalharmonics dependent upon said digitized 3D wave field signals acquiredby said first order sensors; and constructing a 3D wave fieldrepresentation dependent upon said computed even and odd coefficientsfor said acquired 3D wave field signals.
 18. The computer readablestorage medium according to claim 17, wherein the constructed 3D wavefield representation is used for any one of the following applications:active noise cancellation; beamforming; direction of arrival estimation;and sound recording or reproduction.
 19. The method according to claim1, wherein said acquired 3D wave field signals are any one of acoustic,radio frequency wave, and microwave.
 20. The apparatus according toclaim 8, wherein said acquired 3D wave field signals are any one ofacoustic, radio frequency wave, and microwave.
 21. The computer readablestorage medium according to claim 17, wherein said acquired 3D wavefield signals are any one of acoustic, radio frequency wave, andmicrowave.